Finite Elements in Analysis and Design 31 (1999) 273 — 280 Application of Zienkiewicz—Zhu’s error estimate with superconvergent patch recovery to hierarchical p-reﬁnement Hyung-Seok Oh, R.C. Batra* Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219, USA Abstract The Zienkiewicz—Zhu error estimate is slightly modiﬁed for the hierarchical p-reﬁnement, and is then applied to three plane elastostatic problems to demonstrate its eﬀectiveness. In each case, the error decreases rapidly with an increase in the number of degrees of freedom. Thus Zienkiewicz—Zhu’s error estimate can be used in the hp-reﬁnement of ﬁnite element meshes.  1999 Elsevier Science B.V. All rights reserved. Keywords: Adaptive p-reﬁnement; Error estimate; Superconvergent patch recovery 1. Introduction One way to control the quality of a ﬁnite element solution with an optimal use of computational resources is to reﬁne the mesh adaptively. The adaptive ﬁnite element analysis generally consists of two stages: a posteriori error estimate and the mesh reﬁnement. The goal is to reﬁne the mesh so that the error is within the speciﬁed tolerance and is as uniformly distributed throughout the domain as possible. Two types of a posteriori error estimates, namely the post-processing and the residual, have been employed. The post-processing type error estimate was proposed for the h-reﬁnement by Zienkiewicz and Zhu [1]. They used the nodal averaging method to obtain recovered stresses and compared stresses interpolated from the recovered stresses with those computed from the ﬁnite element solution at the quadrature points to ﬁnd the error in the numerical solution. Henceforth, we will refer to this error estimate as the Zienkiewicz—Zhu’s (Z) error estimate. Many authors [2—5] have shown its eﬀectiveness in the h- and r-reﬁnements. The nodal averaging procedure or the ¸  projection technique for obtaining recovered stresses is valid * Corresponding author. Fax: #1 540 231 4574; e-mail: rbatra@vt.edu 0168-874X/99/$ — see front matter  1999 Elsevier Science B.V. All rights reserved PII: S 0 1 6 8 - 8 7 4 X ( 9 8 ) 0 0 0 6 3 - 8